Only slightly related: anyone know how to make porcelain or other normal tiles? If I wanted to redo the bathroom in these or whatever, can they be made?
A chiral aperiodic monotile
21–30 of 83 posts
Re: A chiral aperiodic monotile
#22Only slightly related: anyone know how to make porcelain or other normal tiles? If I wanted to redo the bathroom in these or whatever, can they be made?
You need pretty strict tolerances on the dimensions of this Spectre tile, I think. Normal Penrose tiles might be easier.
That said I have no idea in the first place how you'd get your hands on bespoke tiles...
Re: A chiral aperiodic monotile
#23Only slightly related: anyone know how to make porcelain or other normal tiles? If I wanted to redo the bathroom in these or whatever, can they be made?
They're either usually formed in a mold by pressing. The resulting "green" tile is sometimes stamped or patterned, and glaze is applied. Then they are fired in a kiln.
Re: A chiral aperiodic monotile
#24Only slightly related: anyone know how to make porcelain or other normal tiles? If I wanted to redo the bathroom in these or whatever, can they be made?
You need pretty strict tolerances on the dimensions of this Spectre tile, I think. Normal Penrose tiles might be easier. That said I have no idea in the first place how you'd get your hands on bespoke tiles...
The grout when tiling allows for a fair amount of fudging things, though from experience exactly dimensioned tiles are MUCH easier to work with.
Re: A chiral aperiodic monotile
#25How could you address each tile, to create an aperiodic tile map? Would be a neat tech demo. Like HyperRogue, for example. https://www.roguetemple.com/z/hyper/
Re: A chiral aperiodic monotile
#26This excites me much more than the original result, which I considered to use two tiles[1]. The fact that it’s a such tiny modification of the original result is crazy. Even if you don’t intend to read the paper, look at the illustrations of the hierarchical substitution algorithm at the top of pages 6 and 7, those are just beautiful.
[1] The authors discuss various historic definitions of tilings and whether reflections should be allowed or not (they argue that most definitions allow them). For me, the answer is simple: nature is chiral, you can’t reflect things willy-nilly. Puzzle pieces, bathroom tiles, even polygons in 3D rendering all have distinguishable sides.
Re: A chiral aperiodic monotile
#27Can anyone explain why people are interested in tilings?
Earthquake resistance of buildings. See Incan usage of aperiodic masonry to spread out the frequency response of the construction.
Similar principle with Apple's laptop fan blades.
Same mechanism might be responsible for the Boson peak phenomena in amorphous materials and quasicrystals, where the macro-structure creates extra capacity for absorbing lower-than-lattice-frequency vibrations than what the crystal-structure alone predicts.
It's all about Fourier analysis.
Re: A chiral aperiodic monotile
#28Earlier quoted context omitted.
One of the things this has consequences on are the physics of crystals and pseudocrystals.
Hmmm... now that we know what shapes these tiles take, maybe we can look for or design molecules with the "same shape" (and yes I know molecules aren't 2D planar objects, but some can be approximated as such, e.g. the benzene ring) and with the right molecule-to-molecule attractions, such that they naturally arrange themselves into these aperiodic tilings.
TRANSPARENT ALUMINUM!?
Without a periodic crystal structure, it is likely to transmit light (see glass's transparency due to its amorphousness)
Re: A chiral aperiodic monotile
#29Hmmm... I wonder if these spectres are potentially a basis for a new form of cryptographic algorithms. Unique non-repeating sequences exclusively derived from a set of rules and an initial state in multiple dimensions sounds like a promising candidate.
I'm not understanding why that's different than seeded random sequence or the sequential digits of any irrational number? The latter guarantees a non-repeating sequence and can be trivially generated with square roots.
The question is in reversibility. A typical PRNG is hard to exactly reverse, but easy to brute-force, given a few values from the sequence.
I think it may be way harder to brute-force a tiling, because the number of tiles grows quadratically relative to the distance from an initial configuration. I wonder how easy would it be to step back.
Re: A chiral aperiodic monotile
#30Earlier quoted context omitted.
I'm not understanding why that's different than seeded random sequence or the sequential digits of any irrational number? The latter guarantees a non-repeating sequence and can be trivially generated with square roots.
The question is in reversibility. A typical PRNG is hard to exactly reverse, but easy to brute-force, given a few values from the sequence. I think it may be way harder to brute-force a tiling, because the number of tiles grows quadratically relative to the distance from an initial configuration. I wonder how easy would it be to step back.
I think lattices are probably more secure though. It's a similar idea to tiling but you can run it in a huge number of dimensions, without having to change the underlying math.